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Exercises Sequences, counting and probability

Example 2 from Algebra and Trigonometry, 13.4 Series and Their Notations

Finding the First n Terms of an Arithmetic Series

Find the sum of each arithmetic series.

ⓐ $5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32$
ⓑ $20 + 15 + 10 +…+ −50$
ⓒ $\sum_{k = 1}^{12} 3 k - 8$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
1. ⓐ
  We are given $a_{1} = 5$ and $a_{n} = 32.$
  Count the number of terms in the sequence to find $n = 10.$
  Substitute values for $a_{1}, a_{n},$ and $n$ into the formula and simplify.
  $\begin{array}{l} \begin{array}{l} S_{n} = \frac{n (a_{1} + a_{n})}{2} \end{array} \\ S_{10} = \frac{10 (5 + 32)}{2} = 185 \end{array}$
2. ⓑ
  We are given $a_{1} = 20$ and $a_{n} = - 50.$
  Use the formula for the general term of an arithmetic sequence to find $n .$
  $\begin{array}{l} a_{n} = a_{1} + (n - 1) d \\ - 50 = 20 + (n - 1) (- 5) \\ - 70 = (n - 1) (- 5) \\ 14 = n - 1 \\ 15 = n \end{array}$
  Substitute values for $a_{1}, a_{n}, n$ into the formula and simplify.
  $\begin{array}{l} \begin{array}{l} S_{n} = \frac{n (a_{1} + a_{n})}{2} \end{array} \\ S_{15} = \frac{15 (20 - 50)}{2} = - 225 \end{array}$
3. ⓒ
  To find $a_{1},$ substitute $k = 1$ into the given explicit formula.
  $\begin{array}{l} a_{k} = 3 k - 8 \\ a_{1}=3(1)-8=-5 \end{array}$
  
  We are given that $n = 12.$ To find $a_{12},$ substitute $k = 12$ into the given explicit formula.
  $\begin{array}{l} a_{k}=3k-8 \\ a_{12} = 3 (12) - 8 = 28 \end{array}$
  Substitute values for $a_{1}, a_{n},$ and $n$ into the formula and simplify.
  $\begin{array}{l} S_{n}=\frac{n (a_{1} + a_{n})}{2} \\ S_{12} = \frac{12 (- 5 + 28)}{2} = 138 \end{array}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

$1 .4 + 1 .6 + 1 .8 + 2 .0 + 2 .2 + 2 .4 + 2 .6 + 2 .8 + 3 .0 + 3 .2 + 3 .4$

How did it go?

PreviousUsing Summation Notation Next Solving Application Problems with Arithmetic Series

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 13.4, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.